Assignment_6

# Assignment_6 - What boundary conditions are implied by this...

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ME 303 ADVANCED ENGINEERING MATHEMATICS Assignment 6 Question 1 . Consider a rectangular membrane of density ρ , thickness h, and dimensions a x 0 , b y 0 . The membrane is held fixed around its perimeter and is stretched to a tensile stress of T σ . By applying Newton’s second law to a y x Δ Δ element and then shrinking the element to a point (i.e. 0 Δ x , 0 Δ y ), derive the 2-D wave equation for the membrane displacement ( ) t y x s , , , perpendicular to the () y x , plane. Question 2. The result of Question 1 should be the following PDE: 2 2 2 2 2 2 2 + = y s x s c t s with ρ T = 2 c (a) Verify by substitution into the PDE that a solution is + = t b a c b y a x K t y x s 2 2 1 1 cos sin sin ) , , ( π where K is a constant. (b)
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Unformatted text preview: What boundary conditions are implied by this solution? (c) What initial conditions on displacement and velocity are implied by this solution? Question 3. Consider heat conduction in a flat, circular disc of thickness h . The disc is insulated on both its top and bottom surfaces, such that heat can flow only in the ( ) θ , r plane (using polar coordinates). By applying conservation of energy to an element of sides r Δ , Δ r , and thickness h , show that the disc temperature is governed by the 2-D diffusion equation in polar coordinates: ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ 2 2 2 2 2 1 1 T r r T r r T a t T where c k a =...
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